Generating Markov evolutionary matrices for a given branch length (Q1938705)

Under a Markovian evolutionary process, the expected number of substitutions per site (branch length) that occur when a sequence evolves from another via a transition matrix \(P\) can be approximated by \(-\frac 14 \log(\det P)\). In continuous-time models, it is easy to simulate the process for any given branch length. For discrete-time models, it is not so trivial. In this article, the authors solve this problem for the most well-known discrete-time models JC69*, K80*, K81*, SSM, and GMM. They also provide concise algorithms to generate stochastic matrices of given determinant. These models have the advantage to be nonhomogeneous evolutionary processes.